Purpose This paper is motivated by a simple question: Does the satisfaction of friendship-and-love differ from the satisfaction of consumption of substantive goods such as clothing and shelter? The answer of standard economics is straightforward: all satisfactions can be reduced to a common metric, called “utility,” “wellbeing” or “welfare.” Most social scientists and nonstandard economists disagree. They maintain that the two satisfactions are incommensurable. However, such scientists generally fail to pinpoint exactly what makes the two genera of satisfaction incommensurable. This paper aims to pinpoint the difference between the two genera of satisfaction with the aid of Adam Smith’s moral theory. Design/methodology/approach This paper’s method relies on a close reading of Smith’s Theory of Moral Sentiments (TMS). Indeed, it focuses on a short chapter at the outset of TMS, where Smith identifies what he calls “mutual sympathy” as the source of the satisfaction of friendship-and-love. Findings Adam Smith stumbled on what this paper calls the “paradox of friendship-and-love”: Given that fellow-feelings mirror the original emotions, why does the sharing of a sad event with a friend rather generate the opposite, joy? To solve this paradox, Smith distinguishes between everyday satisfaction, what economists call “wellbeing” and what this paper calls “substantive utility,” on the one hand, and the joy of friendship-and-love, what this paper calls “transcendent utility,” on the other hand. One’s transcendent feeling is always pleasant, i.e. irrespective of the substrate event. This “always” pleasant feature of transcendent feeling sets friendship-and-love apart from substantive utility. Research limitations/implications The proposed solution to the paradox has a theoretical implication. Namely, the distinction between two genera of satisfaction entails corresponding distinction between two genera of approval/disapproval that is pertinent to business ethics: i) informed by substantive satifaction, the first genus is the approval of honest choice (i.e. rational) and disapproval of dishonest choice (i.e. nonrational); and ii) informed by transcendent satisfaction, the second genus is the approval of sincere behavior, which does not manipulate friendship for an ulterior motive, or the disapproval of insincerity. Practical implications The proposed solution to the paradox has a practical implication. This solution allows us to understand taboos that prohibit the commodification of goods – such as taboos prohibiting the buying-and-selling of human kidneys, votes and sex. Such taboos simply prohibit the conflation or substitution between substantive satisfaction and the satisfaction of friendship-and-love. The existence of taboos should prove the incommensurability thesis regarding the two genera of satisfaction. Originality/value This paper offers a new solution to the paradox of friendship. This paper offers a new interpretation of Smith’s moral theory relying on rational choice theory.

Understanding susceptibility to online influence is crucial for mitigating the spread of misinformation and protecting vulnerable audiences. This paper investigates susceptibility to influence within social networks, focusing on the differential effects of influence-driven versus spontaneous behaviors on user content adoption. Our analysis reveals that influence-driven adoption exhibits high homophily, indicating that individuals prone to influence often connect with similarly susceptible peers, thereby reinforcing peer influence dynamics, whereas spontaneous adoption shows significant but lower homophily. Additionally, we extend the Generalized Friendship Paradox to influence-driven behaviors, demonstrating that users' friends are generally more susceptible to influence than the users themselves, de facto establishing the notion of Susceptibility Paradox in online social influence. This pattern does not hold for spontaneous behaviors, where friends exhibit fewer spontaneous adoptions. We find that susceptibility to influence can be predicted using friends' susceptibility alone, while predicting spontaneous adoption requires additional features, such as user metadata. These findings highlight the complex interplay between user engagement and characteristics in spontaneous content adoption. Our results provide new insights into social influence mechanisms and offer implications for designing more effective moderation strategies to protect vulnerable audiences.

The friendship paradox, initially discussed by Scott Feld in 1991, highlights a counterintuitive social phenomenon where individuals tend to have fewer friends than their friends do on average. The sociological implications of this paradox are profound, as it can create a distorted understanding of social norms and consequently influence beliefs, attitudes, and behaviors, particularly when highly connected individuals present a skewed representation of those norms. In essence, it can lead individuals to misjudge what is typical or desirable within their social circles. This study investigates the temporal dynamics of the friendship paradox using smartphone communication data from over 600 incoming freshmen at the University of Notre Dame participating in the NetHealth project. By tracking the friendship index– the ratio of an individual’s friends’ average number of friends to their own number of friends– over 119 days during the Fall semester of 2015, we examine how the paradox evolves over time. Our findings reveal that the friendship index stabilizes more rapidly than both the individuals’ own degree and the variation among their friends’ degrees. Results from the latent growth-curve model (LGCM) confirm that while the friendship index continues to increase, its growth rate declines over time. Moreover, the LGCM identifies individual degrees, ethnic backgrounds, and personality traits as influential factors shaping the manifestation and development of the friendship paradox. By exploring the mechanisms underlying this paradox in a dynamic communication network, this study enhances our understanding of the structural factors influencing the evolution of the friendship paradox in digitally mediated interactions.

Abstract Let $$G_n$$ G n be an undirected finite graph on $$n\in {\mathbb {N}}$$ n ∈ N vertices labelled by $$[n] = \{1,\ldots ,n\}$$ [ n ] = { 1 , … , n } . For $$i \in [n]$$ i ∈ [ n ] , let $$\Delta _{i,n}$$ Δ i , n be the friendship bias of vertex i, defined as the difference between the average degree of the neighbours of vertex i and the degree of vertex i itself when i is not isolated, and zero when i is isolated. Let $$\mu _n$$ μ n denote the friendship-bias empirical distribution, i.e., the measure that puts mass $$\frac{1}{n}$$ 1 n at each $$\Delta _{i,n}$$ Δ i , n , $$i \in [n]$$ i ∈ [ n ] . The friendship paradox says that $$\int _{\mathbb {R}}x\mu _n(\textrm{d}x) \ge 0$$ ∫ R x μ n ( d x ) ≥ 0 , with equality if and only if in each connected component of $$G_n$$ G n all the degrees are the same. We show that if $$(G_n)_{n\in {\mathbb {N}}}$$ ( G n ) n ∈ N is a sequence of sparse random graphs that converges to a rooted random tree in the sense of convergence locally in probability, then $$\mu _n$$ μ n converges weakly to a limiting measure $$\mu $$ μ that is expressible in terms of the law of the rooted random tree. We study $$\mu $$ μ for four classes of sparse random graphs: the homogeneous Erdős-Rényi random graph, the inhomogeneous Erdős-Rényi random graph, the configuration model and the preferential attachment model. In particular, we compute the first two moments of $$\mu $$ μ , identify the right tail of $$\mu $$ μ , and argue that $$\mu ([0,\infty ))\ge \tfrac{1}{2}$$ μ ( [ 0 , ∞ ) ) ≥ 1 2 , a property we refer to as friendship paradox significance.

A central concern of community ecology is the interdependence between interaction strengths and the underlying structure of the network upon which species interact. In this work we present a solvable example of such a feedback mechanism in a generalized Lotka-Volterra dynamical system. Beginning with a community of species interacting on a network with arbitrary degree distribution, we provide an analytical framework from which properties of the eventual "surviving community" can be derived. We find that highly connected species are less likely to survive than their poorly connected counterparts, which skews the eventual degree distribution towards a preponderance of species with lower degrees. Furthermore, the average abundance of the neighbors of a species in the surviving community is lower than the community average (reminiscent of the famed friendship paradox). Finally, we show that correlations emerge between the connectivity of a species and its interactions with its neighbors. More precisely, we find that highly connected species tend to benefit from their neighbors more than their neighbors benefit from them. These correlations are not present in the initial pool of species and are a result of the dynamics.

Modeling, analysis and validation of friendship paradox in evolving networks

The friendship paradox states that, on average, people have fewer friends than their friends do. This can lead to social perception biases, such as the belief that one’s friends are more socially engaged than oneself. In a recent paper, we investigated the consequences of this type of social comparison for content sharing behavior in online social networks (Medhat and Iyer, The Friendship paradox and social network participation, international workshop on complex networks and their applications, pp 301–315, 2023). We simulated a scenario where people compare the feedback that their own content receives to the feedback that their friends’ content receives and adjust their sharing based on the comparison. If sharing and feedback rates are initially uniform over individuals, then feedback disparities initially depend solely upon the local structural friending paradox. These structurally induced disparities may then induce sharing disparities that further amplify (or potentially, reduce) feedback disparities, triggering cycles of updates of sharing rates and feedback disparities. In our previous work, we observed that monotonic responses to social comparisons, where larger disparities result in greater behavioral changes, led to an asymptotic decline in overall network sharing. In this paper, we extend our earlier work by studying fundamental properties of our friendship-paradox-induced sharing trajectories (FIT) method. We analyze how sharing rates trend when it is dictated by both observed feedback disparities and activity thresholds for collective behavior. We also evaluate the iterative impact of FIT on network nodes, as a measure of node centrality in a network, finding it to have the ability to identify central nodes at a community level and to do so at a lower computational complexity than community detection, a property not matched by commonly used node centrality methods such as betweeness, closeness and degree centrality.

A critical element of word of mouth (WOM) or buzz marketing is to identify seeds, often central actors with high degree in the social network. Seed identification typically requires data on the relevant network structure, which is often unavailable. We examine the impact of WOM seeding strategies motivated by the friendship paradox, which can obtain more central nodes without knowing network structure. Higher degree nodes may be less effective as seeds if these nodes communicate less with neighbors or are less persuasive when they communicate; therefore, whether friendship paradox–motivated seeding strategies increase or reduce WOM and adoption remains an empirical question. We develop and estimate a model of WOM and adoption using data on microfinance adoption across village social networks in India. Counterfactuals show that the proposed strategies with limited seeds are about 13%–30% more effective in increasing adoption relative to random seeding. These strategies are also on average 5%–11% more effective than the firm’s leader seeding strategy. We also find these strategies are relatively more effective when we have fewer seeds. This paper was accepted by Juanjuan Zhang, marketing. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2024.4991 .

Estimating exposure to information on a social network is a problem with important consequences for our society. The exposure estimation problem involves finding the fraction of people on the network who have been exposed to a piece of information (e.g., a URL of a news article on Facebook, a hashtag on Twitter). The exact value of exposure to a piece of information is determined by two features: the structure of the underlying social network and the set of people who shared the piece of information. Often, both features are not publicly available (i.e., access to the two features is limited only to the internal administrators of the platform) and are difficult to estimate from data. As a solution, we propose two methods to estimate the exposure to a piece of information in an unbiased manner: a vanilla method that is based on sampling the network uniformly and a method that non-uniformly samples the network motivated by the Friendship Paradox. We provide theoretical results that characterize the conditions (in terms of properties of the network and the piece of information) under which one method outperforms the other. Further, we outline extensions of the proposed methods to dynamic information cascades (where the exposure needs to be tracked in real time). We demonstrate the practical feasibility of the proposed methods via experiments on multiple synthetic and real-world datasets.

Asymptotic distribution of the friendship paradox of a random geometric graph

We provide the mathematical and empirical foundations of the friendship paradox in networks, often stated as "Your friends have more friends than you." We prove a set of network properties on friends of friends and characterize the concepts of ego-based and alter-based means. We propose a network property called inversity that quantifies the imbalance in degrees across edges and prove that the sign of inversity determines the ordering between ego-based or alter-based means for any network, with implications for interventions. Network intervention problems like immunization benefit from using highly connected nodes. We characterize two intervention strategies based on the friendship paradox to obtain such nodes, with the alter-based and ego-based strategy. Both strategies provide provably guaranteed improvements for any network structure with variation in node degrees. We demonstrate that the proposed strategies obtain several-fold improvement (100-fold in some networks) in node degree relative to a random benchmark, for both generated and real networks. We evaluate how inversity informs which strategy works better based on network topology and show how network aggregation can alter inversity. We illustrate how the strategies can be used to control contagion of an epidemic spreading across a set of village networks, finding that these strategies require far fewer nodes to be immunized (less than 50%, relative to random). The interventions do not require knowledge of network structure, are privacy-sensitive, are flexible for time-sensitive action, and only require selected nodes to nominate network neighbors.

The Friendship Paradox is a simple and powerful statement about node degrees in a graph. However, it only applies to undirected graphs with no edge weights, and the only node characteristic it concerns is degree. Since many social networks are more complex than that, it is useful to generalize this phenomenon, if possible, and a number of papers have proposed different generalizations. Here, we unify these generalizations in a common framework, retaining the focus on undirected graphs and allowing for weighted edges and for numeric node attributes other than degree to be considered, since this extension allows for a clean characterization and links to the original concepts most naturally. While the original Friendship Paradox and the Weighted Friendship Paradox hold for all graphs, considering non-degree attributes actually makes the extensions fail around 50% of the time, given random attribute assignment. We provide simple correlation-based rules to see whether an attribute-based version of the paradox holds. In addition to theory, our simulation and data results show how all the concepts can be applied to synthetic and real networks. Where applicable, we draw connections to prior work to make this an accessible and comprehensive paper that lets one understand the math behind the Friendship Paradox and its basic extensions.

The “friendship paradox” of social networks states that, on average, “your friends have more friends than you do”. Here, we theoretically and empirically explore a related and overlooked paradox we refer to as the “enmity paradox”. We use empirical data from 24,678 people living in 176 villages in rural Honduras. We empirically show that, for a real negative undirected network (created by symmetrizing antagonistic interactions), the paradox exists as it does in the positive world. Specifically, a person’s enemies have more enemies, on average, than a person does. Furthermore, in a mixed world of positive and negative ties, we study the conditions for the existence of the paradox, which we refer to as the “mixed-world paradox”, both theoretically and empirically, finding that, for instance, a person’s friends typically have more enemies than a person does. We also confirm the “generalized” enmity paradox for non-topological attributes in real data, analogous to the generalized friendship paradox (e.g., the claim that a person’s enemies are richer, on average, than a person is). As a consequence, the naturally occurring variance in the degree distribution of both friendship and antagonism in social networks can skew people’s perceptions of the social world.

It is often of interest to sample vertices from a graph with a bias towards higher-degree vertices. One well-known method, which we call random neighbor or RN, involves taking a vertex at random and exchanging it for one of its neighbors. Loosely inspired by the friendship paradox, the method is predicated on the fact that the expected degree of the neighbor is greater than or equal to the expected degree of the initial vertex. Another method that is actually perfectly analogous to the friendship paradox is random edge, or RE, where an edge is sampled at random, and then one of the two endpoint vertices is selected at random. Obviously, random sampling is only required when full knowledge of the graph is unattainable. But, while it is true in most cases that knowledge of all vertices’ degrees cannot be obtained, it is often trivial to learn the degree of specific vertices that have already been isolated. In light of this, we suggest a tweak to both RN and RE, inclusive random sampling. In inclusive random neighbor (IRN) the initial vertex and the selected neighbor are considered, in inclusive random edge (IRE) the two endpoint vertices are, and in both cases, we learn the degree of each and select the vertex of higher degree. This paper explores inclusive random sampling through theoretical analysis and experimentation. We establish meaningful bounds on IRN and IRE’s performances, in particular in comparison to each other and to their exclusive counterparts. Our analyses highlight differences of the original, exclusive versions as well. The results provide practical insight for strategizing a random sampling method, and also highlight graph characteristics that impact the question of which methods will perform strongly in which graphs.

Abstract In this paper, we consider the friendship paradox in the context of random walks and paths. Among our results, we give an equality connecting long-range degree correlation, degree variability, and the degree-wise effect of additional steps for a random walk on a graph. Random paths are also considered, as well as applications to acquaintance sampling in the context of core-periphery structure.

The Friendship Paradox—the principle that “your friends have more friends than you do”—is a combinatorial fact about degrees in a graph; but given that many web-based social activities are correlated with a user’s degree, this fact has been taken more broadly to suggest the empirical principle that “your friends are also more active than you are.” This Generalized Friendship Paradox, the notion that any attribute positively correlated with degree obeys the Friendship Paradox, has been established mathematically in a network-level version that essentially aggregates uniformly over all the edges of a network. Here we show, however, that the natural node-based version of the Generalized Friendship Paradox—which aggregates over nodes, not edges—may fail, even for degree-attribute correlations approaching 1. Whether this version holds depends not only on degree-attribute correlations, but also on the underlying network structure and thus can’t be said to be a universal phenomenon. We establish both positive and negative results for this node-based version of the Generalized Friendship Paradox and consider its implications for social-network data.

Abstract The Friendship Paradox is a phenomenon which states that most people have fewer friends than their own friends, and its generalization has been proposed in the last three decades by several scientific papers. Our study is focused on the academic environment, and seeks to determine whether or not the impression that individuals may have concerning invitations to take part in oral defenses is justifiable. This involved testing two hypotheses with regard to academic committee members: “The Invitee Paradox” (in terms of the person who is invited); and “The Inviter Paradox” (in terms of the person who extends the invitation). The paradoxes were assessed by designing invitation networks, both weighted and unweighted, which represent a dual relationship in which an invitation originates from an “inviter” and is extended to an “invitee”. We then tested the hypotheses with the aid of two real-world open access datasets from online academic repositories: (1) American (Brazilian Capes Catalog); and (2) European (French STAR Deposit). Our results showed that only “The Invitee Paradox” was true. We also explored possible relations between our proposed measurement of the invitation paradoxes and the PageRank metric, as to evaluate the relative importance of members in the invitation networks.

A heterogeneous structure of social networks induces various intriguing phenomena. One of them is the friendship paradox, which states that on average, your friends have more friends than you do. Its generalization, called the generalized friendship paradox (GFP), states that on average, your friends have higher attributes than yours. Despite successful demonstrations of the GFP by empirical analyses and numerical simulations, analytical, rigorous understanding of the GFP has been largely unexplored. Recently, an analytical solution for the probability that the GFP holds for an individual in a network with correlated attributes was obtained using the copula method but by assuming a locally tree structure of the underlying network [Jo et al., Phys. Rev. E 104, 054301 (2021)]. Considering the abundant triangles in most social networks, we employ a vine copula method to incorporate the attribute correlation structure between neighbors of a focal individual in addition to the correlation between the focal individual and its neighbors. Our analytical approach helps us rigorously understand the GFP in more general networks, such as clustered networks and other related interesting phenomena in social networks.

The Friendship Paradox in the Formation of Academic Committees

Contact Tracing for Disease Containment: a Network-Based Analysis*